For continuous-time flows, the possibility of this “Lyapunov function”—with its distinctive property \(\dot{\Phi }<0\)  outside the attractors—was suggested in the context of the general stability problem of dynamical systems [1]} and in a sense, it adds a quantitative dimension to the qualitative theory of differential equations. The linearization of the flow around its attractors always provides such a function, but its validity breaks down well inside their own basin. Instead, finding a global Lyapunov function is not an easy problem. If only the information of the (deterministic) dynamical system is to be used, this function can be found for the so-called “gradient flows”—purely irrotational flows in 3d, exact (longitudinal) forms in any dimensionality. But since for general relaxational flows (having nontrivial Helmholtz–Hodge decomposition) the integrability conditions are not automatically met, some more information is needed.
